TITLE: Homotopy groups of $MO(2)$
QUESTION [13 upvotes]: Have there been any computations of the higher homotopy groups of $MO(2)$, the Thom space of the universal $O(2)$-bundle? Thom himself noted in his landmark 1954 paper that
$$
\pi_1(MO(2))=0,\quad \pi_2(MO(2))=\mathbb{Z}/2,\quad\pi_3(MO(2))=0,\quad \pi_4(MO(2))=\mathbb{Z}.
$$
By the Pontrjagin-Thom construction $\pi_n(MO(2))$ is the group of cobordism classes of embeddings of closed $(n-2)$-manifolds in $\mathbb{R}^n$, where a cobordism is an embedding in $\mathbb{R}^n\times [0,1]$.

REPLY [6 votes]: There are a couple of references which show that $\pi_5(MO(2))=\mathbb{Z}/2$.
Suzuki, H., On the realization of the Stiefel-Whitney characteristic classes by submanifolds, Tohoku Math. J., II. Ser. 10, 91-115 (1958). ZBL0107.17001.
The proof uses the Postnikov tower. The conclusion is on page 108.
Zvagel’skij, M. Yu., Cobordisms of embeddings with codimension two, J. Math. Sci., New York 104, No. 4, 1276-1282 (2001); translation from Zap. Nauchn. Semin. POMI 252, 40-51 (2001). ZBL1052.57042.
The proof here is geometric, and (for me) harder to follow. This paper also contains a proof that $\pi_6(MO(2))=0$.